Numerical Study of Tip Vortex Cavitation
Transcript of Numerical Study of Tip Vortex Cavitation
Numerical Study of Tip Vortex Cavitation
SHF Workshop : Hydraulic Machinery, Cavitation.
3-4 June 2015, Nantes (CETIM), France
Dr. Jean Decaix* and Pr. Cecile Munch, Univ. of Applied Sciences and Arts - WesternSwitzerland Valais, Sion, SwitzerlandPr. Guillaume Balarac, Univ. Grenoble Alpes, LEGI, CNRS, F38000, Grenoble, FranceDr. Matthieu Dreyer and Pr. Mohamed Farhat, Ecole Polytechnique Federale de Lausanne,Laboratory for Hydraulic Machines, Lausanne, Switzerland
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CONTENTS
1. CONTEXT
2. OBJECTIVES
3. TEST CASE
4. MODELLING
5. NUMERICAL SET UP
6. RESULTS
7. CONCLUSION
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CONTEXT
HYDRONET 2
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TIP VORTEX
Tip vortex in Kaplan turbine
• Promotes cavitation.
• Flow topology of the tip-leakageflow ?
• Gap width influence ?
• Vortex control ?
• Scale up from the model to theprototype ?
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OBJECTIVES
COUPLING EXPERIMENTAL AND NUMERICAL INVESTIGATION
• To investigate the gap width influence.
• To investigate the flow topology in the gap.
• To investigate the cavitation influence.
Induced vortex
Tip-leakage vortex
Tip-separation vortex
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TEST CASE : NACA0009
Main characteristics
• Chord length : c = 0.1 m.
• Inlet velocity : U∞ = 10 m/s.
• Reynolds number : Rec = 106.
• Cross section : 1.5c × 1.5c .
• Gap width : 0.1c .
• Blade incidence : 10.
• Cavitation number : σexpe = 2.1, σnum = 1.3.
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MODELLING : Turbulence
Reynolds Averaged Navier-Stokes (RANS) equationsContinuity and momentum equations :
∂ρ
∂t+
∂ρui∂xi
= 0
∂ρui∂t
+∂ρuiuj∂xj
= −∂p
∂xi+
∂ (σij + τij)
∂xj
With :
σij = ρν
(∂ui∂xj
+∂uj∂xi
)
; τij = −ρu′
i u′
j = ρνt
(∂ui∂xj
+∂uj∂xi
)
νt computed using the k − ω SST model.
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MODELLING : Cavitation
Homogeneous Relaxation Model (HRM)Mixture density defined as :
ρ = αLρL + (1− αL) ρV
Transport equation for αL :
∂αL
∂t+ uj
∂αL
∂xj= mv +mc
With (Kunz’s model [1]) :
mv =ρ
ρL
Cv
t∞
min (p − pvap , 0)
0.5 ρLU2∞
mc =ρ
ρL
Cc
t∞α2L
max (p − pvap , 0)
max (p − pvap , 0.01pvap)
[1] R. F. Kunz et al., ”A preconditioned Navier - Stokes method for two-phase Flows with application to cavitation prediction”,Comput. Fluids, vol. 29, pp. 849-875, 2000.
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NUMERICAL DOMAIN
Flow direction
Gap
Computational domain
• Length L = 8c : 2c upstreamand 5c downstream.
• Section : 1.5c × 1.5c .
Structured mesh
• 2 millions of nodes.
• 30 nodes in the gap.
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NUMERICAL SET UPOpenFOAM 2.1.0 : interPhaseChangeFoam algorithm
Boundary conditions
• No slip wall.
• Inlet : normal velocity u∞ = 10 m/s.
• Outlet : gradient free.
• Pressure reference sets in one cell.
Numerical Schemes
• Time scheme : second order (”backward”) with ∆ t = 10−5 s.
• Convective scheme : second order based on the Total Variation Diminishing (TVD)approach (”limitedLinear” and ”vanLeer”).
• Laplacian scheme : second order (”linear corrected”).
• Sharp interface is preseved introducing a counter-gradient in the transport equationfor αL.
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VORTEX CORE POSITIONwithout cavitation
x/c = 1x/c = 1.2
x /c= 1.5y
x
z
Pitchwise positionx/c y/c (Expe) y/c (RANS)1 0.14 0.131.2 0.18 0.181.5 0.30 0.22
Spanwise positionx/c z/c (Expe) z/c (RANS)1 0.12 0.101.2 0.13 0.131.5 0.16 0.15
More detailled results in :
1. J. Decaix, G. Balarac, M. Dreyer, M. Farhat, and C. Munch, 2015, ”RANS and LES simulations of the tip leakage vortex fordifferent gap widths”, Journal of Turbulence, Volume 16, Issue 4, pp. 309-341
2. M. Dreyer, J. Decaix, C. Munch, M. Farhat, 2014, ”Mind the gap : a new insight into the tip leakage vortex usingstereo-PIV”, Experiments in Fluids, Volume 55, Issue 11
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AXIAL VORTICITYwithout cavitation
Position x/c = 1
Experiment RANS Computation
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VELOCITY FIELD
Dimensionless streamwise velocity component in the mid-plane
Withoutcavitation
Withcavitation
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CAVITATING TIP VORTEX
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VORTEX CORE ?
Q =1
2
(
||Ω||2 − ||S||2)
x/c = -0.3
x/c = -0.1x/c = 0.1
x/c = 0.3x/c = 0.5
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VORTICITY EQUATION
∂~ω
∂t+ (~u.∇)~ω = (~ω.∇)~u − ~ω∇.~u + ν∇2~ω + νt∇
2~ω +1
ρ2∇ρ ∧∇p
︸ ︷︷ ︸
RHS
Cross plane at x/c = 0
Without cavitation With cavitation
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CONCLUSION
Non-cavitating tip vortex is captured accurately compared to theexperiment.
Vorticity at the vortex core is under-estimated
⇒ Pressure drop at the vortex core is also under-estimated.⇒ Cavitating computations are performed at a lower cavitation number
(σnum = 1.3 instead of σexpe = 2.1).
Cavitating tip vortex shows a qualitative agreement with the experiment.
Vortex core identification depends on the criterion chosen : Maximum of the void fraction. Maximum of the Q-criterion.
The production of streamwise vorticity is negative in case of cavitation.
A detail analysis is expected in an uncoming paper.
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THANK YOU FOR YOUR
ATTENTION
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